When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-space

endowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and let

The purpose of this note is to prove that the Jacobian of these systems is given by the equivalence

where Kh is Eger's operator, defined as follows:

In this paper we apply pointwise subsequence ergodic theory to show that certain sequences of integers constructed randomly are multiply intersective. Analogous results for sequences constructed from Hardy fields are also indicated.

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

An approximate formula is found for the field produced on the boundary of an obstacle in an inhomogeneous medium by a source which is close to the boundary. Both harmonic and aperiodic waves are discussed.

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

In the last two decades a family of closely related functions have been introduced for the solution of quantum-mechanical problems. In dealing with atomic and molecular systems Pauling (12) introduced in 1928 linear combinations of the eigenfunctions of a single atom, which were called hybrids. These have specific symmetry properties which were utilized by Kimball (6) to construct a group-theoretical method for their determination. More recently Lennard-Jones (8) defined linear combinations of molecular eigenfunctions which he called equivalent orbitals. These are formally similar to the hybrids and Lennard-Jones took over the Kimball method for their determination.

For a handlebody H, we define two graphs, the augmented disk graph $\mathcal{ADG}(H)$ and the truncated augmented disk graph $\mathcal{TADG}(H)$, and we show they are hyperbolic in the sense of Gromov. In the process, we show they are quasi-isometric to two other disk graphs defined by U. Hamenstädt, the super conducting disk graph $\mathcal{SDG}(H)$ and the electrified disk graph $\mathcal{EDG}(H)$ respectively. So we reprove two theorems of Hamenstädt [12].

Our approach uses techniques from Masur–Schleimer's study on the hyperbolicity of the disk graph $\mathcal{DG}(H)$ [21].

1. It often happens that the greater parts of two variables x, y may be connected linearly but there may be other disturbances of both. A treatment given by one of us ((1), section 4·42, pp. 216–17) assumes that the other disturbances are independent, as for observational errors. This is not altogether satisfactory because unless the variations of xr, yr are much greater than the standard errors there is an asymmetry in the treatment corresponding to the well-known distinction between the two regression lines.

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curve

which satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

1. A topological game is a game in which the positions are the points of a topological space; the set of possible moves from any such point varies continuously with the point. Play can start from any point of the space and at each point it is specified which of the players has the initiative. Play ends when a position is encountered at which the set of positions from which the player with the initiative can choose is empty. The payoff to each player depends on the set of positions met in the play. The notion of topological game is due to Berge (1), (2).

We show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n, is given by c q n −1 + O(n −2). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m. To that end, we first derive an improved estimate for the proportion of polynomials of degree n, all of whose non-constant divisors have degree greater than m. In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permutations.

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.

Let G be a nilpotent, connected and simply connected real Lie group and let Γ be a lattice in G. A necessary and sufficient condition is given for a Lie subgroup of G to act ergodically on Γ[setmn ]G. As an application, we characterize the irreducible unitary representations of G for which the restriction to Γ remains irreducible.